Electron Spin

In filling the k-space of a metal with electrons, each grid point in the k-space can be occupied with two electrons, one in a spin up state and one in a spin down state. Energy of the electron does not depend on the spin state of the electron. The full quantum state of the electron then includes the position wavefunction and the spin orientation. This can be notates as follows:

The spin of a particle can also be represented as a two-element matrix or spinor, with spin up represented as (1;0) and spin down as (0;1).

http://hyperphysics.phy-astr.gsu.edu/hbase/spin.html

The Sommerfeld Model – Metals

Following the Drude model describing the movement of electrons in metals, Sommerfeld developed yet another model for electrons in metals in 1927. This new model would account for electron energy distributions in metals, Pauli’s exclusion principle, and Fermi-Dirac statistics of electrons. This model factors in quantum mechanics and the Schrödinger Equation.

The Somerfeld model’s view of electrons in metals can be taken as an example of a large volume of metal with electrons confined in the volume. We’ll call this a potential well. Inside this volume or potential well, electrons are ‘free’ with zero potential. Outside the potential well, the potential is infinite. The electron states inside this box are governed by the Schroedinger equation.

The quantum state of an electron is generally described by the Schroedinger equation as shown below.

The result of the Schroedinger equation, applying the boundary conditions of the problem with the potential being zero at the boundary, the solution of the wavefunction is below. This solution introduces a concept called the k-space, a 3D grid of allowed quantum states.

The density of grid points in the k-space is related to Lx, Ly, and Lz of the solution. The number of grid points per unit volume or density (i.e. density of states) will be V/(2*pi)^3, where the spacing of points in the 3D k-space are defined as 2pi/Lx, 2pi/Ly, and 2pi/Lz.

Drude Model – Metals

In 1900, Paul Drude formed a theory of conduction in metals using the newly discovered concept of the electron. The theory states:

1. Metals have a high density of free electrons.
2. Electrons in metals move according to Newton’s laws
3. Electrons in metals scatter when encountering defects, ions and the momentum of the electron is random after this scattering event.

In short, the Drude Model explains how electrons can be expected to move in metals, which is fundamental to the operation of many devices.

Applied Electric Field

In the presence of an electric field, electron motion on average can be described with the following momentum p(t), when tau is the scattering time and 1/tau is the scattering rate:

Now consider several cases for electron motion in metals.

No applied electric field, E(t) = 0
In this case, electrons move randomly and the electron path momentum averages zero.

Constant Uniform Electric Field
When a uniform electric field is present, electron movement averages in the opposite direction of the electric field.

Relating momentum to velocity, we can find the electron drift velocity which is the rate at which the electron travels in an average direction caused by the field. The electron mobility is the relationship between drift velocity and the electric field.

Electron current density is related to the number of electrons, an electron’s charge, the mobility, and the electric field. The factor between electron current density and electric field is the conductivity.

Nanofabrication Processes

Due to the size of the structures made during nanofabrication, the semiconductor wafers are highly vulnerable to foreign objects, such as dust. The waveguide structures being fabricated see a width of 2 microns. Interference by a piece of dust of similar width results in device fabrication failure if left on the wafer before depositing an oxide layer. For this reason, cleanrooms and their procedures are designed to eliminate dust particles and organic material that may invade the device fabrication process. Still, caution must be taken at each step to ensure that there is no dust on one’s sample. The wafer should be observed under a microscope before each etching and depositing procedure and cleaned using the appropriate method, depending on the previous and next steps.

If the wafer does not have a layer of photoresist on it, the cleaning procedure is to dip the wafer into a beaker of acetone, then isopropanol and then DI water. The typical duration of this is for about 1 minute each. If there is a layer of old photoresist on the wafer, this may need to be increased to five minutes each. If old photoresist remains, a flood UV exposure using the mask aligner and development may proceed a second solvent wash. The RIE machine is also used for removing photoresist on wafers if used on an O2 descum process.

Masks are templates used in the mask aligner to create patterns on wafers. These plates must also be cleaned after using, since they will make contact with wafers that have photoresist on them. This is especially critical for masks that have small patterns (<20 microns). After finishing use, soak in acetone, isopropanol and DI water for 5 minutes or longer. If after inspecting the mask on a microscope there is still photoresist remaining, an ultrasonic bath for 20 minutes or more or a flood exposure on the contact side of the mask can be attempted, however it likely will not be needed if it is cleaned continually after use for each day.

If the wafer being used requires a custom epitaxial structure, the first step in the fabrication process is to grow these layers. A common method for research applications is molecular beam epitaxy (MBE). MBE machines are used primarily for research due to their accuracy but slow growth rate. MOCVD, on the other hand is used for simpler epitaxial structures and mass production.

PECVD is used to deposit oxides such as SiO2, SiNx and others. If using small wafers, using a larger carrier wafer is common practice. When creating a PECVD deposit recipe, the gas mixture and temperature are selected.

The E-beam evaporator is used to deposit metals such as gold, aluminum, chrome, platinum and other materials such as germanium.

F E-Beam Evaporator

Reactive Ion Etching (RIE) is used for etching oxides and deposits on wafers using a chemical plasma that is charged with an electromagnetic field and under a strong vacuum. This is termed a dry etching process. For RIE etching recipes, the pressure, chemical and RF power are chosen. The RIE is also used to remove photoresist and organic material using an O2 clean process.

The ICP (or Inductively Coupled Plasma) tool can etch many materials including SiO2, SiNx, Cr, GaAs and AlGaAs. ICP etch recipes are designed using a selected pressure, RF and ICP power, etchant gas and temperature. When using the ICP, run a chamber cleaning process with O2 and Argon with a dummy wafer loaded. After a cleaning run, the desired etch process should be run with a dummy wafer first before loading the desired wafer. For smaller wafers, thermal conducting paste can be applied between the wafer and a larger carrier wafer. For deep wafer etches on semiconductor material such as GaAs, the edges of the wafer will be etched more. To avoid this, other wafers can be placed aside it.

Acid Etching is a wet etch process, unlike RIE and ICP. This is performed at a bench using a blend of chemicals to etch the semiconductor wafer itself, typically. Heavier protective gear is worn during this process to prevent contact with some of the most dangerous chemicals used in a nanofabrication facility. Hydroflouric acid, a deadly neurotoxin is one chemical that is used frequently in a nanofabrication facility [37]. One use of an acid etch recipe used is an AlGaAs-selective etch.

Photolithography is a technique used in semiconductor device fabrication. First, a light-sensitive layer called photoresist is added to a semiconductor wafer. Depending on the type of resist (positive or negative), this layer can be removed using developer after applying UV light. A mask is template used to apply UV light only to a desired region or shape on the wafer. After this is done, etching can be performed exclusively to the parts of the wafer without a photo-sensitive layer.

The spinner is the machine that is used to apply photoresist to a wafer. The wafer is first held on a vacuum arm and photoresist is applied. The vacuum arm is then spun at a desired spin rate and duration. This creates a uniform film of photoresist on the wafer. After running on the spinner, the wafer should sit on a hot plate for a specified time and temperature.

Particularly for non-circular wafers, photoresist can build up along the edges, creating an uneven surface. This is problematic for the following steps, so the photoresist is removed from the edges and underside of the wafer using a swab and acetone.

If there is already a pattern on the wafer, the wafer position in the mask aligner can be adjusted to ensure alignment. It is recommended to include an alignment feature on the mask die, such as a veneer mark, especially if alignment is critical to that fabrication step. After aligning as needed, the UV light exposure time is selected and applied to the wafer. The wafer is dipped in developer for a specified time, then in DI water, and gently blow dried with a nitrogen gun.

Lift-off photoresist is used when creating a metal feature on a wafer. In this process, normal photoresist can be applied over the lift-off resist (before putting on a hot plate). After running the wafer in the spinner, lift-off resist needs to be removed from the edges using tweezers. Lift-off resist will react differently to acetone, so for edge bead removal, a separate solution needs to be used. After performing photolithography and metal depositions, the lift-off resist may need to be developed using yet another type of solution. For LOR 20-3, it is recommended that the wafer sit on a hot plate at 80 degrees C in the lift-off developer solution for 12 hours. It also recommends a wash in cool lift-off developer and isopropanol after sitting on the hot plate. Refer to data sheets for specific instructions for chemicals.

Electron beam Lithography (EBL) performs the same role as the mask aligner, but with much higher precision due to the smaller electron wavelength. Photoresist still needs to be applied before using an EBL machine. Instead of using a mask, it follows the pattern on a GDSII file. An EBL can be used for all lithography steps in fact, though it is much slower, so it is used only for steps with a narrow feature that is not achievable on a mask aligner or stepper. An EBL machine also contains an SEM, scanning electron microscope and can be used for that function as well.

The SEM or Scanning Electron Microscope is used for examining structures that are too small for an optical microscope. This is achieved using an electron beam, which excites conductive materials. SEM is necessary for inspecting etch qualities and for making accurate measurements of component sizes. The SEM operates by directing an electron gun at the sample. The charged atoms release electrons, producing the signal that is detected for the image.

One tool that is useful for measuring the height profile on a wafer is the Dektak tool. This is often used after an etching process. This gives the height profile along a line on the wafer.

The ellipsometer is used for measuring the thickness of a film deposit. Unlike the Dektak, the ellipsometer is able to measure multiple layers. However, it is not useful for precise positioning or height profiles over a distance on the wafer. The ellipsometer is typically only used for uniform deposits on wafers that have not been exposed to etching or photolithography, unless the shapes are very large.

Optical Loss in Optical Waveguides and Free Carrier Absorption

Sources of loss in optical waveguides include free carrier absorption, band edge absorption, surface roughness, bending loss, and two photon absorption. Optical loss can be determined from the imaginary index of refraction.

Band edge absorption is a wavelength-dependent absorption based on material properties. For wavelengths above the bandgap wavelength (approx. 1 micron), the band edge absorption and free-carrier absorption of GaAs is greatly reduced. Free-carrier absorption caused by doping is still a concern for optical waveguide loss, however.

Free carrier absorption is loss in optical waveguides due to interaction of photons and charge carriers. The effects of free carrier absorption can be calculated using the free carrier coefficients of electrons and holes for the material and the doping concentration. Since doping is used to create a PIN structure, it is therefore wiser based on free carrier absorption to have the regions surrounding the intrinsic waveguide core to be lightly doped. The imaginary dielectric constant due to free carrier absorption, based on doping levels is calculated as follows. The doping concentration for electrons and holes are n and p respectively, the bulk refractive index is n0, the wavenumber is k, and FCN and FCP are the free carrier coefficients of electrons and holes respectively.

Converting from normalized SFDR (dBHz^(2/3)) to real SFDR (dB)

SFDR is frequently written in the units of dBHz^(2/3), particularly for fiber optic links. Fiber optic links can often have such high bandwidth, that assuming a bandwidth in SFDR is unhelpful or misleading. Normalizing to 1Hz therefore became a standard practice. The units of SFDR for a real system with a bandwidth are dB.

Now consider that the real system has a specific bandwidth. The real SFDR can be calculated using the following formulas:
SFDR_real = SFDR_1Hz – (2/3)*10*log10(BW)

Here are a few examples.

Noise Figure in a Microwave Photonic Link

The standard definition for noise figure (NF) is the degradation of signal to noise ratio (SNR). That is, if the output noise power of the system is increased more than the output signal power, then this implies a significant noise figure and a degredation of SNR.

For an RF photonic link, there are a couple assumptions that result in a slightly altered definition and calculation for noise figure. One assumption is that the input noise is the thermal noise (kT), such as would be detected from an antenna receiver. It is also the case that RF photonic links may be employed in a case where the input signal power level is not defined. In simple telecommunications aplications, it is standard to expect a certain input power level, but as a communications system at a radar front end for instance, the input signal is not known. We can use the gain of the link as a relationship between output signal and input signal instead of a known input and output signal power.

It is a goal of the link designer in those cases to ensure that all true signals can be distinguished from noise. For these reasons, we may also think of noise figure in the following definition:

Noise figure (NF) is the difference between the total equivalent input noise and thermal background noise.

The equivalent input noise is the output noise without considering the gain of the link.

For the noise figure calculation, we have then:

NF = 10*log_10( EIN / GkT ),

where EIN is the equivalent input noise, G is the link gain, k is Boltzmann’s constant, and T is the temperature in Kelvin.

Equivalent input noise (EIN) is as follows:

EIN = GkT + <I^2>*Z_PD,

where <I^2> is the current noise power spectral density at the output of the link and Z_PD is the photodetector termination impedance.

These together, we have noise figure:

NF = 10*log_10(1+(<I^2*Z_PD)/GkT)

Noise Sources in RF Photonic Links

Identifying the noise sources in an RF Photonic link allows us to determine the performance of the link and helps us to identify critical components to link and device design to develop a high performance link. Below is an intensity modulated optical link. Other modulation schemes in RF photonic links may be discussed at a later point.

Since the output of the RF photonic link is the photocurrent generated by the photodetector, the noise sources are a current noise power spectral density.

Noise sources from the laser:

Laser RIN (relative intensity noise) is the fluctuation of optical power. Relative intensity noise is the noise of the optical power divided by the average optical power in a laser. RIN noise originates from spontaneous radiative carrier recombination and photon generation.

Noise sources from the modulator:

Noise in a modulator is due to thermal noise of electrode termination and ohmic loss in the electrodes.

Noise sources from the photodetector:

Shot noise occurs as a result of the quantization of discrete charges or photons. Noise is also generated by the photodetector termination.

Total current noise power spectral density of the RF photonic link:

RF Photonic links (also called Microwave Photonic Links) are systems that transport radiofrequency signals over optical fiber. The essential components of an RF photonic link are the laser as a continuous-wave (CW) carrier, a modulator as a transmitter and the photodetector as a receiver. A low-noise amplifier is often used before the modulator.

Optical fiber boasts much lower loss over longer distances compared to coaxial cable, and this flexibility of optical fiber is one advantage over conventional microwave links. Another advantage of RF photonic links are their immunity to electromagnetic interference, which plays a more significant role in electronic warfare (EW) applications. RF Photonic links are employed in telecommunications, electronic warfare, and quantum information processing applications, although the performance requirement in each of these situations vary. In telecommunications, a high bandwidth is required, while in EW applications having high spurious-free dynamic range (SFDR) and a low noise figure (NF) is critical. In quantum information processing applications, a low insertion loss is critical.

In EW scenarios, unlike in telecommunications, the expected signal frequency and signal power is unknown. This is because typically, an RF photonic link is found as a radar receiver. In a system with high SFDR and low NF, distortion is minimized, the radar has stronger reliability and range, and smaller signals can be registered. Here is a demonstration of two scenarios with different SFDR and NF:

Low SFDR, High NF:

High SFDR, Low NF:

Transformer Circuit Review: Ideal Transformers, Conservation of Energy

In a closed system, energy can be transferred through different forms (heat, kinetic energy, potential energy, etc), but not created nor destroyed. For a passive device such as a transformer, the energy in the system must also follow. This is termed conservation of energy.

A transformer is a passive circuit component which follows these basic formulas, where P is the power and n is the number of turns on the transformer:

Pout=Pin

n2/n1=V2/V1

Consider an electrical transformer with turns ratio N, what is the output voltage?  What is the output current?

The output voltage Vout=N*Vin.

A correct answer to this question must satisfy that power is conserved. This means that the output power must equal the input power. Power = voltage * current.

The impedance of the transformer for N turns ratio:

Zout = Vout/Iout = (N*V­in)/(Iin/N) = N2 * Zin

Thermal Background Noise

Any object with a temperature above absolute zero (Kelvin) radiates electromagnetic energy, or thermal noise. Noise is generated by the earth and cosmos, and this is background thermal noise, which is received by an antenna.

Thermal background noise is the starting point for system performance. A signal of strength below the thermal background noise will be indistinguishable from noise.

The thermal background noise power is proportional to the temperature (P = kTB, k being Boltzmann’s constant, T the temperature in Kelvin, and B the bandwidth in hertz). The thermal background noise power spectral density is the fundamental noise minimum at -174 dBm/Hz at 300K.

The gain of the device or system further amplifies the thermal background noise. RF Photonic links most often use a low noise amplifier (LNA) directly before the modulator, amplifying the thermal background noise.

The definition of thermal noise applied to electronics is the movement of charge carriers caused by temperature in a conductor.

Mean Squared Noise Power

What does it mean when people say “mean squared”?

The average value of a noise waveform is zero. The square of the waveform mean is also equal to zero. The square of the noise signal and the mean of the square are non-zero. This is because the negative values associated with the zero-mean noise waveform are made positive by squaring, and the entire waveform is positive. Taking the root of the averaged square of the waveform yields the RMS.

The mean of the squared (“mean square”) noise waveform is the noise power with respect to a 1 Ohm resistor (units: V2/Ω=W, “power” if noise signal is a voltage signal, and units I2/Ω=W, also “power” if noise waveform is current).

The power spectral density is the power of the signal in a unit bandwidth.

What is a current noise power spectral density?

The correct definition of current noise spectral density is the mean of the squared current per hertz, <i2>. The units are A2/Hz.

The square of the mean is equal to zero, because the mean of the noise waveform is zero and squaring that number remains zero. The mean of the square is a non-zero number. Taking the square of a noise current results in a positive valued current waveform. Taking the average of the square is a non-zero number used for the spectral density.

Why is it that RF waves travel faster than c/√(εμ) in a coplanar waveguide (CPW) electrode?

In electro-optic modulators, one important task is matching the propagating RF and optical wave velocities. This begins a discussion on modulator electrode design.

The primary method of matching the RF and optical velocities is using a slow wave electrode, that is, a capacitively loaded electrode.

Before capacitive loading, we need to determine the initial velocity of RF waves travelling in the coplanar waveguide (CPW). The CPW electrode is as follows. The optical waveguides are positioned between the signal and ground electrodes.

Let’s think about the formula for wave propagation velocity for RF waves:

V_RF = c/n,

where c is the speed of light (3×10^8 m/s) and n is the microwave index. n is also equal to:

n = √(εμ),

where ε and μ are the relative permittivity and permeability of the medium that the RF waves are propagating in.

We might think, if we know what material the modulator is made of, then we can calculate the microwave index based on the relative permittivity and permeability of the material, and calculate it. Not so fast…

As shown in the diagram above, the propagating electromagnetic wave’s mode is not confined within the substrate. For this reason, we must determine a weighted average of the material properties based on the electromagnetic waves’ mediums, namely air and the substrate. I use Ansys HFSS to perform this calculation easily and accurately.

In summary, the propagating electromagnetic waves along the coplanar waveguide electrodes are present both in the semiconductor substrate and in air surrounding the device. The velocity of the propagating electromagnetic wave is therefore a weighted average of the electric field propagation in air and the semiconductor. Since the index of air is less than the semiconductor, the field propagates faster than if it were entirely propagating in semiconductor.

What are the frequencies of the second-order and third-order distortion tones given two frequency peaks?

In general, the third order distortion tones are understood to exist as in-band distortion at frequencies 2ω21 and 2ω12 in a two tone intermodulation test. Third order distortion also exists at frequencies ω1 and ω2. Second order distortion tones are found outside of a narrowband system at 2ω2, 2ω1, and ω12.

Consider the two-tone input of a non-linear system with frequencies ω1 and ω2:

Vin = A[cos(ω1t)+cos(ω2t)]

The second order and third order distortion tones are calculated on the following page. In summary, the tones are shown in the table below. This shows that third order distortion tones are found not only in the positions mentioned above, but also contribute to the fundamental tone frequencies.  In a spurious-free system, all third order tones will be below the noise floor. This is verified in MATLAB with ω1, ω2 at 500kHz, 501kHz.

What does the term “Spurious-free” mean in Spurious-free Dynamic Range (SFDR)?

In the term spurious-free dynamic range (SFDR), spurious-free means that non-linear distortion is below the noise floor for given input levels. The system is spurious when non-linear distortion is present above the noise floor. The system is spurious-free when non-linear distortion is below the noise floor. SFDR therefore is the range of output levels whereby the system is undisturbed by non-linear distortion or spurs.

SFDR contrasts with compression dynamic range (or linear dynamic range (LDR)) which is the range of output levels whereby the fundamental tone is proportional to the input, irrespective of distortion tone levels. The fundamental tone is no longer considered to be linear beyond the 1dB compression point, after which the output fundamental tones do not increase at the same rate as the input fundamental tones.

Spurs are non-linear distortion tones generated by non-linearities of a system. The output of a non-linear system can be modeled as a Fourier series.

The first term a0 is a DC component generated by the non-linear system. The second term a1Vin is the fundamental tone with some level of gain a1. The third term a2Vin2 is a second order non-linear distortion tone. The fourth term a3Vin3 is the third-order non-linear distortion tone. Further expansion of the Fourier series generates more harmonic and distortion tones. Even order harmonic distortion tones are usually outside of the band of interest, unless the system is very wideband. Odd order distortion tones however are found much closer to the fundamental tone in the frequency domain. SFDR is usually taken with respect to the third order intermodulation distortion, however it may also occasionally be taken for the fifth order (or seventh).